Praxis II - 5003 - Mathematics

Composing and Decomposing Numbers

"composing" So 349 is composed of 3 hundreds, 4 tens and 9 ones, in other words: 300 + 40 + 9 ⇒ 349 "Decomposing" Decomposing is when we break the number apart 349 ⇒ 300 + 40 + 9 "Decomposing"

Prime and Composite Numbers

*A prime number has only two factors: 1 and itself. *A composite number has more than two factors. The number 1 is neither prime nor composite. To determine if a number is prime or composite, follow these steps: Find all factors of the number. If the number has only two factors, 1 and itself, then it is prime. If the number has more than two factors, then it is composite.

Using formulas to determine unknown quantities

1.) Fill in the blank: 2 + ? = 10. The answer is 8, since 2 + 8 = 10. This solution may be particularly helpful for someone with 2 fingers on one hand and 8 fingers on the other. 2.) What number doubled equals 24? In other words, if you were to sleep half the day away, how long would that be? In symbols, we know 2 · ☐ = 24, and we need to fill in the box with the appropriate number. In this case, the answer would be 12, since 2 · 12 = 24 Link: https://www.shmoop.com/algebraic-expressions/variables-unknown-quantities.html

Whole Number Exponents

10 ^ 5 = 10 x 10 x 10 x 10 x 10

Lines, Line Segments, and Rays

Line - Extends forever in both directions Line Segment - is just part of a line. It has two endpoints. Rays - starts at one point and continues on forever in one direction. https://www.khanacademy.org/math/basic-geo/basic-geo-lines/lines-rays/a/lines-line-segments-and-rays-review

Rational Numbers

Sometimes when dividing there is something left over. It is called the remainder. Example: There are 7 bones to share with 2 pups. But 7 cannot be divided exactly into 2 groups, so each pup gets 3 bones, and there is 1 left over: Or each pup gets 3.5 bones

The value of a unit fraction decreases as the value of the denominator increases

Unit Fraction = A fraction with a numerator equal to 1 The larger the denominator of a unit fraction is a smaller part of 1 it represents. example - 3 is smaller than 8, 1/3 + 1/3 + 1/3 = 1 8 -- 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 1 the denominator tells the unit* If you are breaking the pie into 8 parts the parts will be smaller

Interpreting solutions of multi-step one-variable linear equations and inequalities

http://study.com/academy/lesson/interpreting-solutions-of-multistep-linear-equations-inequalities.html

Adds and subtracts linear algebraic expressions

2V + 3 + 7v - 1 2 and 7 are rational coefficients 2n + 6 and (15n) - 29 2n + 6 + (-15n) - 29 = 2n + 6 + (-15n) - 29 = 2n + (-15n) + 6 - 29 = -13n - 23 *If it is negative it must remain negative, if it is positive it must remain positive

Evaluating simple algebraic expressions (i.e. one variable, binomial) for given values of variables

4 - 2f when f = 1 9 - 8/S when S=4 http://www.nabla.hr/BA-AlgebraicExpress.htm *** (read up on this)

Differentiating between the dependent and independent variables in a formula

A variable that depends on one or more other variables. For equations such as y = 3x - 2, the dependent variable is y. The value of y depends on the value chosen for x. Example: On your math quiz, you earn 5 points for each question that you answer correctly. In the equation below, x represents the number of questions that you answer correctly on your math quiz, and y represents the total number of points that you score on your quiz. The relationship between these two variables can be expressed by the following equation: y= 5x y=5x A variable that depends on one or more other variables. For equations such as y = 3x - 2, the dependent variable is y. The value of y depends on the value chosen for x. Dependent Variable: Independent Variable: Example: The Adams Family Restaurant has several types of hamburgers on their menu. The number of hamburger buns they use each day depends on the number of hamburgers their customers order. b = the number of hamburger buns used h = the number of hamburgers ordered Which of the variables is independent and which is dependent? Link: https://www.ixl.com/math/algebra-1/identify-independent-and-dependent-variables

Inverse Operations

Additive Inverse Property - when you add a number to its opposite, the result is always 0 2 + -2 = 0 Multiplicative Inverse Property - when you multiply any number by its opposite, the result is always 1 213 x 1/213 = 1 Additive Property - When you add any number to zero the number stays the same 7 + 0 = 7 Multiplicative Property - any time you multiply a number by 1 the number does not change 13 x 1 = 13

Using mathematical terms to identify parts of expressions and describe expressions

Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation. 2x+4y−9 Term: Each expression is made up of terms. A term can be a signed number, a variable, or a constant multiplied by a variable or variables. Factor: Something which is multiplied by something else. A factor can be a number, variable, term, or a longer expression. For example, the expression 7x(y+3) . Coefficient: The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, 2x+4y−9 Constant: A number that cannot change its value. In the expression 2x+4y−9 Like Terms: Terms that contain the same variables such as 2m, 6m, 7xy If an expression has more than one constant terms, those are also like terms. sum of a number and 5 = (n + 5) Difference of a number and 7 = (m - 7) Product of 6 and a number = (6x) Quotient of a number and 9 = (y / 9) https://www.varsitytutors.com/hotmath/hotmath_help/topics/parts-of-an-expression

Composing and Decomposing Fractions

Below is a pan of brownies that Debbie has baked for her birthday party. If 4 kids each ate one brownie, what expression shows the number of brownies that were eaten? 1/8 + 1/8 + 1/8 + 1/8 = 4/8 3/4 = 1/4 + 1/4 + 1/4 https://plus.maths.org/content/unit-fractions

Decimal Numbers

Decomposing: 37.2 ⇒ 30 + 7 + 0.2 Composing: 4 + 0.7 + 0.08 ⇒ 4.78

Finding Factors and Multiples of Numbers

FACTORS: "Factors" are the numbers we can multiply together to get another number: 2 x 3 = 6 (2 and 3 are factors) 3 × 4 = 12, 3 and 4 are factors of 12 2 × 6 = 12, 2 and 6 are factors of 12, 1 × 12 = 12, 1 and 12 are factors of 12 AND because multiplying negatives makes a positive, −1, −2, −3, −4, −6 and −12 are also factors of 12: (−1) × (−12) = 12 (−2) × (−6) = 12 (−3) × (−4) = 12 So ALL the factors of 12 are: 1, 2, 3, 4, 6 and 12 AND −1, −2, −3, −4, −6 and −12 MULTIPLES:

Use the distributive property to generate equivalent linear algebraic expressions

Having an X that is not raised to a power is a signature mark of a linear equation Distributive Property - To remove the parentheses, we multiply each term inside of our parentheses with the value outside of the parentheses.

Applying the concepts of ratios and unit ratios to compare two quantities

Link: https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/e/comparing-rates

The same whole number must be used when COMPARING FRACTIONS

Link: https://www.montereyinstitute.org/courses/DevelopmentalMath/TEXTGROUP-1-8_RESOURCE/U02_L1_T5_text_final.html

Parallel and Perpendicular Lines

Parallel: slopes are the same! y = mx + b Perpendicular: Two lines are Perpendicular if they meet at a right angle (90°). When we multiply their slopes, we get −1

Order of Operations

Please - Parentheses Excuse - Exponents My - Multiplication Dear - Division Aunt - Addition Sally - Subtraction

Translating between verbal statements and algebraic equations

Represent the cost of access to a movie web site which charges a $5 fee for the first month, and then $28 a month for continued access after the first month. Numerical Example: For 4 months the cost would be: Cost = $5 + $28•(4-1). Replace 4 with m for number of months. c = cost, m = months c = 5 + 28(m - 1) If chocolate doughnuts cost $1.10 and chocolate chip muffins cost $2.25, how many of each can you purchase for $20? Numerical Example: The cost of 3 doughnuts and 2 muffins will be: Cost = $1.50•3 + $2.25• 2. Replace 3 with d for doughnuts, 2 with m for muffins, and Cost with $20. d =doughnuts, m = muffins 20 = 1.50d + 2.25m Express the area of a rectangle whose length is twice its width decreased by 6. Numerical Example: If the width is 10, the length is 14, then the area, A = 10 • 14. Just replace 10 with w for width, and 14 with 2w - 6 for the length. A = area, w = width A = w • (2w - 6) Link: https://mathbitsnotebook.com/Algebra1/AlgebraicExpressions/AEtranslations.html

Standard/Expanded Form

Standard and Expanded Form Some people call the two different forms "Standard" and "Expanded": 349 300 + 40 + 9 Standard Form: 349 Expanded Form: 300 + 40 + 9

Commutative, Distributive, and Associative Properties

The "Commutative Laws" say we can swap numbers over and still get the same answer ... Addition: a + b = b + a 6 + 3 = 3 + 6 Multiplication: a * b = b * a 2 x 4 = 4 x 2 The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) Addition: (a + b) + c = a + (b + c) (6 + 3) + 4 = 6 + (3 + 4) (2 + 4) + 5 = 6 + 5 = 11 2 + (4 + 5) = 2 + 9 = 11 Multiplication: (a × b) × c = a × (b × c) (3 × 4) × 5 = 12 × 5 = 60 3 × (4 × 5) = 3 × 20 = 60 "Distributive Property" a × (b + c) = a × b + a × c 3 * (2 + 4) = 3 * 2 + 2 * 3 What is 6 × 204? 6 × 204 = 6×200 + 6×4 = 1,200 + 24 = 1,224 What is 16 × 6 + 16 × 4? 16 × 6 + 16 × 4 = 16 × (6+4) = 16 × 10 = 160 What is 26×3 - 24×3? 26×3 - 24×3 = (26 - 24) × 3 = 2 × 3 = 6 What is 6×7 + 2×7 + 3×7 + 5×7 + 4×7? = (6+2+3+5+4) × 7 = 20 × 7 = 140 https://www.mathsisfun.com/associative-commutative-distributive.html

Differentiating between an algebraic expression and equation

The primary difference between the two is an equals sign. An "equation" has a left side, a right side and an equals sign separating the sides. An "expression," by contrast, doesn't have any "sides" and is simply what the name suggests: An algebraic "expression." Though sometimes it is possible to combine like terms, we are generally not expected to "do" or "solve" anything regarding expressions. For example: 3x - 7 = 2 This is an EQUATION, because it has a left side, a right side, and an = sign separating the two. 3x - 7 This is an EXPRESSION, because there are no "sides" and no = sign.

Percent as rate per 100

We know that a percent is 1 part out of 100. 1 % is equal to or 0.01. Proportions can be used to solve percent problems. part/whole = percent/100 Example 1: First we have to find 30% of 350. Because the whole is 350 and the percentage is 30%. Step 3) Let n represent the part. Write a proportion and find the cross products to solve for n. 100 n = 30 x 350 100 n = 10,500 100 n ÷ 100 = 10,500 ÷ 100 n = 105 So, Jolly got $105 back from her friend. Example 2: First we have to find 20% of 200. Let r represent the part. Write a proportion and find the cross products to solve for n. 100 r = 20 x 200 100 r = 4000 100 r ÷ 100 = 4000 ÷ 100 r = 40 Step 3) Now we have to find the cost of a bag. Cost - discount = New Price 200 - 40 = 160 So, a customer can buy a bag for $160. Example 3: How many bananas do we have? 1 dozen = 12 bananas 4 dozen = 12 x 4 = 48 4 dozen = 48 bananas Step 3) What percentage of the bananas are gone? Total percentage gone = percent to custard + percent to his son 50%= 40%+10% We have to find 50% of 60. Let r represent the part. Write a proportion and find the cross products to solve for n. 100 r = 50 x 48 100 r = 2,400 100 r ÷ 100 = 2,400 ÷ 100 r = 24 So, he has 24 bananas left.

Examples where multiplication does not result in a product greater than both factors:

Whenever you multiply a positive number by a positive factor less than 1, the product will be smaller than the original number. For example, 1/2 * 3/4 = 3/8 We want to buy 30 roses which are sold in bunches of 5, so we ask for "6 of the 5-rose bunches". In this way, the word times also often means of. If we try using the word of when times appears to have an unclear meaning, we get 11 of 8 rather than 22 times 8. Indeed we know what 1 2 of 8 means - namely 4. (−4) × 8 = -32 1.) Multiplication of a positive number by a number greater than 1 always increases the number. 2.) Multiplication of a positive number by a positive number between 0 and 1 always increases the number. 3.) Multiplication of a negative number by a positive number always increases the first number. 1.) Always True 2.) Always false, as multiplication of a positive number by a number between 0 and 1 will always reduce the number. (e.g. 1 × 12 = 6, 1 × 12 = 4, etc.) 23 3.) Sometimes false and sometimes true; e.g. for the number - 8, 2 × (−8) = −16, so the number is decreased, whereas the number increases in the example below: 1 × (− 8) = − 4 http://www.counton.org/resources/misconceptions/pdfs/miscon02.pdf

Rectangular Arrays Area Models Equations

http://www.familymathnight.com/pdf/RectangularArrays.pdf https://nrich.maths.org/2466 A box of apples has 6 rows of apples with 4 apples in each row. How many apples are in the box? Area Models: Example - A grocery store owner ordered 43 boxes of donuts to sell in her bakery section of her store. There are 24 donuts in each box. How many donuts will she receive in total? 40 3 20 4 40 x 20 = 800 40 x 4 = 160 20 x 3 = 60 4 x 3 = 12 = 1,032 http://mashupmath.com/blog/2017/3/29/teach-your-kids-to-multiply-using-area-models Equations: https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-intro-equations/a/introduction-to-equations

Good Resource for Links to Kahn Academy

https://www.ets.org/s/praxis/pdf/khan_academy.pdf http://www.mathsisfun.com/links/core-grade-5.html

Converting between fractions, decimals, and percents

https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-percent-decimal-conversions/a/converting-between-percents-fractions-decimals

Solving Unit-Rate Problems

https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/e/rate_problems_0.5 http://virtualnerd.com/pre-algebra/ratios-proportions/rates-word-problem-solution.php

Classifying angles based on their measure

https://www.mathplanet.com/education/geometry/points,-lines,-planes-and-angles/measure-and-classify-an-angle

Whole Numbers and Integers

https://www.mathsisfun.com/whole-numbers.html

Composing and Decomposing 2 and 3 dimensional shapes

https://www.thatquiz.org/tq/previewtest?P/D/V/A/68881326065277

Solving multistep one-variable linear equations and inequalities

−2(x+3)<10 −2x−6<10 −2x−6+6<10+6 −2x<16 −2x−2>16−2 x>−8 https://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-linear-inequalities